Question

Three horses \[{H_1},{H_2},{H_3}\] entered a field which has seven portions marked \[{P_1},{P_2},{P_3},{P_4},{P_5},{P_6}\]and $ {P_7} $ . If no two horses are allowed to enter the same portion of the field, in how many ways can the horses graze the grass of the field?

Answer 1

Hint: Find the number of possible ways to graze for $ {H_1} $ . Then find the number of remaining possible ways for $ {H_2} $ . Then do the same for $ {H_3} $ . Product of all three would give the final answer.

Complete step-by-step answer:
There are Seven portions marked \[{P_1},{P_2},{P_3},{P_4},{P_5},{P_6}\] and $ {P_7} $ .
Now, let us say Horse $ {H_1} $ is allowed to start grazing first.
Then Horse $ {H_1} $ can choose to graze in any of the seven available positions.
So, Horse $ {H_1} $ Have $ 7 $ possible ways to graze.
Let us say horse $ {H_1} $ started grazing in position $ {P_1} $ .
Now, Horse $ {H_2} $ is allowed to graze.
But Horse $ {H_2} $ cannot go to portion $ {P_1} $ as horse $ {H_1} $ is already grazing there, and two horses are not allowed to graze simultaneously in the same portion.
Therefore, Horse $ {H_2} $ can graze in any portion from $ {P_2} $ to $ {P_7} $ .
Thus, Horse $ {H_2} $ has $ 6 $ possible ways to graze.
Let Horse $ {H_2} $ is grazing in portion $ {P_2} $ .
Now Horse $ {H_3} $ is allowed to graze horse $ {H_3} $ cannot graze in portion $ {P_1} $ and portion $ {P_2} $ . Because,
Horse $ {H_1} $ is grazing in $ {P_1} $ and Horse $ {H_2} $ is grazing in $ {P_2} $ .
Therefore, Horse $ {H_3} $ has portions $ {P_3} $ to $ {P_7} $ to choose to graze.
Thus, Horse $ {H_3} $ has 5 possible ways to graze.
Therefore, the total number of ways for the horses to graze in field is
 $ = 7 \times 6 \times 5 $
 $ = 210 $ ways
Thus, the horses can graze in $ 210 $

Note: Since we had to find the total possible ways of $ {H_1} $ and $ {H_2} $ and $ {H_3} $ , We used multiplication. If we had to find the number of ways of grazing for any one of $ {H_1} $ or $ {H_2} $ or $ {H_3} $ then we would have used addition.
In permutations, AND represents multiplication and OR represents addition